Discrete Mathematical Structures by Pearson

12-12 Discrete Mathematical Structures

12.4 CALCULUS OF SUBSPACES

THEOREM 12.5 The intersection of any two subspaces of a vector space is a subspace.

Proof: Let W

1

and W

2

be any two subspaces of a vector space V( F ). Let





ab

,

∈∩WW

12

and a,

b ∈ F.

Then



aaa∈∩ ⇒∈ ∈

WW

WW

12

12

and

and



bbb∈∩ ⇒∈ ∈

WW

WW

12

12

and

Now









ab ab∈∈⇒+∈

WWabW

11 1

,

and









ab ab∈∈⇒+∈

WWabW

22 2

,

Hence, ab

WW





ab+∈∩

12

, which shows that W

1

∩ W

2

is a subspace by Theorem 12.4.

Note 12.4

1. It can be observed that W

1

∩ W

2

is the largest subspace contained in W

1

and W

2

as it con-

tains every subspace contained in W

1

and W

2

. The theorem can be generalized and we can

say that the intersection of any family of subspaces of a vector space is a subspace.

2. The union of two spaces ...